To determine whether the statement "The number [tex]\(\sqrt{3}\)[/tex] goes on forever with no repeating pattern; therefore, it is rational" is true or false, let's analyze the properties of the number [tex]\(\sqrt{3}\)[/tex].
1. Definition of Rational and Irrational Numbers:
- A rational number can be expressed as the quotient of two integers, i.e., [tex]\( \frac{p}{q} \)[/tex] where [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are integers and [tex]\( q \neq 0 \)[/tex]. The decimal expansion of a rational number either terminates (comes to an end) or repeats a specific pattern.
- An irrational number cannot be expressed as the quotient of two integers. Its decimal expansion goes on forever without repeating a pattern.
2. Nature of [tex]\(\sqrt{3}\)[/tex]:
- The number [tex]\(\sqrt{3}\)[/tex] cannot be expressed as a fraction of two integers. This makes [tex]\(\sqrt{3}\)[/tex] an irrational number.
- As an irrational number, [tex]\(\sqrt{3}\)[/tex] has a non-terminating and non-repeating decimal expansion. This means that the digits in the decimal representation of [tex]\(\sqrt{3}\)[/tex] will continue infinitely without forming a repeating pattern.
Given the properties discussed, we can assess the statement: "The number [tex]\(\sqrt{3}\)[/tex] goes on forever with no repeating pattern; therefore, it is rational."
- The statement correctly identifies that the decimal representation of [tex]\(\sqrt{3}\)[/tex] goes on forever with no repeating pattern.
- However, it incorrectly concludes that this makes [tex]\(\sqrt{3}\)[/tex] rational. In fact, a number that goes on forever with no repeating pattern is irrational.
Therefore, the correct answer to the question:
"The number [tex]\(\sqrt{3}\)[/tex] goes on forever with no repeating pattern; therefore, it is rational."
is:
B. False
